When a person falls off the shot they are not in good timing or has too much upper body movement during the approach. One drill you can work on to help with balance is the one step drill. Check out Bryan O'Keefe explaining how to do it: https://youtu.be/QZ23LKBLemw . This will help you keep your balance at the line and make better shots. The best thing is that you can work on it at home without a bowling ball.

post your shot,make it a priority. it all starts with footwork and timing to make sure you post your shot. you can change timing with the ball height in your stance and how fast or slow you start with your first step. you also want to push the ball out and forward in the "push away" not up and chase your ball a little always making an effort to post your shot in the end.

Watch his swing.  The feet are going to do what they have to so you don't fall on your face.  If his swing is bumping out and he's trying to project the ball right he is going to fall off every time.